I remember a TV show where one of the characters tells another person that God promised that there would be prophets in each generation. Is there Biblical support for such a view? If this view does exist what are some of the Bible verses that support? I'm interested in the main Catholic and Protestant view on this issue.
A prophet is?
A delegate or mouthpiece of God - cf. PROPHETS AND PROPHECY | Jewish Encyclopedia.
From the said Jewish Encyclopedia above, 'prophecy was not regarded as confined to Israel.'
And from this selected answer | Mi Yodeya
Once the last prophets -- Haggai, Zechariah and Malachi -- died, the prophetic spirit ceased in Israel.
Therefore, at least from this answer, Israel who had Moses, have not had the prophetic spirit in the generations after the last three prophets mentioned above.
In the New Testament St. Paul makes it clear that Jesus gifts his Church with prophets for building up the body of Christ which is the Church
Eph 4:11-14 (RSVCE) 11 And his gifts were that some should be apostles, some prophets, some evangelists, some pastors and teachers, 12 for the equipment of the saints, for the work of ministry, for building up the body of Christ, 13 until we all attain to the unity of the faith and of the knowledge of the Son of God, to mature manhood, to the measure of the stature of the fulness of Christ; 14 so that we may no longer be children, tossed to and fro and carried about with every wind of doctrine, by the cunning of men, by their craftiness in deceitful wiles.
And the Catholic Church teaches that all the baptized share in the priesthood of Christ, in his prophetic and royal mission
CCC 1268 The baptized have become "living stones" to be "built into a spiritual house, to be a holy priesthood."1 By Baptism they share in the priesthood of Christ, in his prophetic and royal mission. They are "a chosen race, a royal priesthood, a holy nation, God's own people, that [they] may declare the wonderful deeds of him who called [them] out of darkness into his marvelous light."2 Baptism gives a share in the common priesthood of all believers.
One can safely conclude that the baptized here on C.SE are prophets (mouthpieces of God) provided their contributions are faithful to the Church's Teaching.